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In this installment on the history of atom theory, physics professor (and my dad) Dean Zollman discusses the work of Louis de Broglie, who sought to explain why electrons adhere to certain orbits. As a grad student, de Broglie looked to the properties of waves. And even Albert Einstein weighed in. – Kim

By Dean Zollman

Dean ZollmanTo create a model that worked, Niels Bohr needed to assume that the electron could orbit the nucleus at only certain radii and that all other radii were forbidden. In the two previous posts, I mentioned that this assumption was a major concern. Bohr could not justify the assumption but used it in order to make the model match the data had been collected for more than half a century.

A justification was forthcoming in the 1920s. In this post, I will discuss this research. However, the underlying reason for the discrete orbits very quickly became much more than just a justification for Bohr’s model. It was the beginning of a revolution in the understanding of atoms and other small objects. And it basically put the Bohr model “out of business” as a serious contender for the model of atoms. More about that revolution will come next few posts. First we need to learn about the critical step to get the revolution started.

From his name alone, you know that Louis-Victor-Pierre-Raymond, seventh duke of Broglie (1892–1987) came from an aristocratic French family. In the 1920s, he was not yet the seventh duke and used the somewhat less pretentious name Louis de Broglie. As a student, he began graduate studies in history but decided that theoretical physics was more interesting. He was motivated in part by his elder brother, the sixth duke of Broglie, an experimental physicist.

Louis de Broglie

Louis de Broglie in 1929 (by Nobel Foundation, public domain, via Wikimedia Commons)

“I was attracted to theoretical physics by the mystery enshrouding the structure of matter and the structure of radiations, a mystery which deepened as the strange quantum concept introduced by Planck in 1900 in his research on black-body radiation continued to encroach on the whole domain of physics,” he said during his Nobel lecture.

For his PhD dissertation, de Broglie was greatly influenced by the work of Planck and Einstein. A few months ago, I discussed how Einstein had explained the photoelectric effect by assuming that light sometimes acted as if it came in small packets (photons) of energy rather than in waves. One of Einstein’s conclusions was that the energy of the photon was equal to Planck’s constant times its frequency. In symbols:

Einstein's equation

The relation between the energy and frequency of a photon. (From The Fascination of Physics by Jacqueline Spears and Dean Zollman, used with permission.)

By applying Planck’s hypothesis, Einstein concluded that light which is normally considered a wave sometimes acted like a particle with its energy given by the equation above.

De Broglie thought that if light could sometimes act as particles, then maybe particles could sometimes act as waves. If objects like electrons behaved like waves, they needed to have a wavelength. Using Einstein’s equation and his special theory of relativity, de Broglie derived an equation for the wave length of any object that has a mass. It says the wavelength is inversely proportional to the mass times the velocity (the momentum). In symbols:

The relation among the wavelength, mass and velocity of an object with mass. (From The Fascination of Physics by Jacqueline Spears and Dean Zollman, used with permission.)

The relation among the wavelength, mass and velocity of an object with mass. (From The Fascination of Physics by Jacqueline Spears and Dean Zollman, used with permission.)

In those days, it was not uncommon for a graduate student to work on his or her own and deliver a completed dissertation to the examining committee. Basically, de Broglie worked in this way. However, his hypothesis was not received well by the committee. Certainly part of the problem that the committee saw was that de Broglie could not refer to a single experiment that supported his work.

A member of the committee was Paul Langevin (1872–1946), a prominent physicist in the 1920s. De Broglie later said that Langevin was “probably a bit stunned by the novelty of my ideas.” Even though he may have been stunned, Langevin sent a copy of the dissertation to Einstein for his review. Einstein rather quickly responded, “I believe it is a first feeble ray of light on this worst of our physics enigmas.” De Broglie wrote in the German translation of his dissertation, “Einstein from the beginning has supported my thesis.” (De Broglie’s dissertation is also available in a modern translation into English at the Fondation Louis de Broglie site. However, it has some heavy duty mathematics.) With Einstein’s endorsement, de Broglie received his PhD.

To explain why the electron in an atom could have only certain orbits, de Broglie relied on the interference property of waves. As shown in the figure below when two waves meet, the result is a bigger wave or a smaller one. On the left side of the figure is the result when the troughs and the crests of the two waves match up. The result shown at the top is a bigger wave. This effect is called constructive interference. On the right, the crests of one wave matches with the troughs of the second. As shown at the top of this one, the two waves cancel each other. This one is destructive interference.

Interference of two waves

Interference of two waves (by Haade, GFDL or CC-BY-SA-3.0, via Wikimedia Commons

For the electrons in atoms, the wave needs to exist as they move in a circle. The drawing below show different situations for these circular waves. For the waves in (a) each of the waves can fit on a circle and close on themselves. When they come back around they act like the waves on the left in the figure above. Toughs meet troughs and crests meet crests, and we have constructive interference. So they add nicely. Because the wave of the electron is not zero, we can conclude that the electron can exist in these two orbits.

However, when the wave in (b) comes back around the troughs will eventually meet crests. They act like the waves on the right of the figure above. Thus, the wave is canceled (destructive interference), so the electron cannot exist in the orbit because its wave is zero. The net result is that only certain radii can meet the condition that an electron’s wave just fits nicely on the circumference of the circle. Only when the electron’s wave can exist, can the electron be in that orbit. Thus, de Broglie’s electron waves give a reason for Bohr’s orbits.

de Broglie's waves in circles

De Broglie waves on a circle. Part (a) shows two waves which can fit on a circle and close on themselves. Part (b) shows one that does not. In (b) when the wave comes back around the troughs and crests meet. The result will be no wave of this wavelength can exist in a circle with that radius. (From The Fascination of Physics by Jacqueline Spears and Dean Zollman, used with permission)

A result of a detailed analysis of the electron orbits and several other phenomena provided evidence that de Broglie was on to something. However, the explanations of the Bohr orbits and other phenomena were indirect evidence of the wave behavior of matter. Constructive and destructive interference can only occur with waves; particles cannot do it. If electrons really have wave properties, physicists should be able to devise an experiment in which they can directly observe the constructive and destructive interference of electrons. Within a few years of publication of de Broglie’s dissertation such experiments were completed on both sides of the Atlantic. There are some interesting stories connected to the experiments, so I will save them for next time.

Dean Zollman is university distinguished professor of physics at Kansas State University where he has been a faculty member for more than 40 years. During his career he has received four major awards — the American Association of Physics Teachers’ Oersted Medal (2014), the National Science Foundation Director’s Award for Distinguished Teacher Scholars (2004), the Carnegie Foundation for the Advancement of Teaching Doctoral University Professor of the Year (1996), and AAPT’s Robert A. Millikan Medal (1995). His present research concentrates on the teaching and learning of physics and on science teacher preparation.


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